noise transmission along shock-waves - Deep Blue - University of

128 Pages·2011·10.14 MB·English

NOISE TRANSMISSION ALONG
SHOCK-WAVES
by
Prasanna Amur Varadarajan
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Aerospace Engineering)
in The University of Michigan
2011
Doctoral Committee:
Professor Philip L. Roe, Chair
Professor Bram van Leer
Professor Smadar Karni
Assistant Professor Krzysztof J. Fidkowski
⃝c Prasanna Amur Varadarajan 2011
All Rights Reserved
To my parents, guru and all my Teachers
ii
ACKNOWLEDGEMENTS
My deepest gratitude to my parents V.Varadarajan and Kumari Varadarajan, for
their love and support throughout my life. Their motivation towards education has
got me this far to complete my doctoral work. I am extremely thankful to my guru
Usha Srinivasan in her conviction towards her work and help me shape as a sensible
individual. I owe a lot to my brother, cousins and all my family members for their
good wishes and support.
At University of Michigan, I am extremely grateful to my advisor Professor Phil
Roe for his guidance throughout my work. I have enjoyed all the conversations I had
with him for the past five years. His patience in letting me understand and explore
things has helped me a lot. It is the amount of freedom he has given me in learning
and making me really think has helped me shape up this work. It is a privilege to be
his student. I would like to thank his wife Jacqueline Roe for her motherly affection
and care she has shown towards me.
I have learned a lot from the courses offered by Professor van Leer and Professor
Karni and I am thankful to them for being in my committee and helping me out with
their valuable suggestions for my thesis work. Special thanks to Professor Fidkowski
forservinginmycommitteeandforhisvaluableinputstomythesis. Anoteofthanks
to all the teachers that I have had in my life who have motivated me to move forward.
Special thanks to Dr Del Vecchio1 of the Electrical department to have supported me
with my research in the field of Bio-medical engineering during my Master’s degree
1currently Associate professor at Department of Mechanical Engineering, MIT
iii
as well in-between during my doctoral work. I would like to thank my undergraduate
advisor Dr. Abdusamad Salih 2 at NIT, Trichy, India whose teaching has been a big
motivation for me to continue my study in the subject of fluid dynamics.
I am thankful to Dr Sergio Pirozzoli of C.F.P.R group, Italy for sharing their
DNS data with us and Dr Farhad Jaberi’s group from Michigan state University for
confirming and validating the results of the numerical Euler computations performed
in this work.
In Michigan over the stay of five years I have acquired a number of wonderful
friends whose company I would cherish for a long time. Many thanks to them and to
my colleagues Daniel Zaide and Tim Eymann for their support. All my friends have
celebrated the success of my doctoral degree as their own. I am greatly indebted to
them all.
Finally, great appreciation to the Department of Aerospace Engineering, Univer-
sity of Michigan, CRASH and MACCAS for providing the financial support, without
which this work would have been not possible.
2currently Assistant professor at Department of Aerospace Engineering, IIST, India
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . xiv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
CHAPTER
I. Background and Motivation . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Geometrical shock dynamics(GSD) . . . . . . . . . . . . . . . 2
1.3 Shock Boundary Layer Interaction(SBLI) . . . . . . . . . . . 4
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II. GSD - Shock propagation in 2D . . . . . . . . . . . . . . . . . . 10
2.1 Brief History of Geometrical shock dynamics(GSD) . . . . . . 10
2.2 Geometrical shock dynamics(GSD) - Governing equations . . 11
2.3 Numerical aspects of Geometrical shock dynamics(GSD) . . . 14
2.4 The A-M Relationship . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 The original proposal . . . . . . . . . . . . . . . . . 16
2.4.2 A modification . . . . . . . . . . . . . . . . . . . . . 17
2.5 Test problems for illustration . . . . . . . . . . . . . . . . . . 21
2.5.1 Shock-diffraction problem - Simple wave solution . . 21
2.5.2 Shock-shock problem - propagation of a shock over a
compression ramp . . . . . . . . . . . . . . . . . . . 24
2.6 Extension of Geometrical shock dynamics(GSD) for a shock
propagating into a moving medium . . . . . . . . . . . . . . . 25
v
2.6.1 Governing equations . . . . . . . . . . . . . . . . . . 25
2.7 Propagation of a sinusoidal shock . . . . . . . . . . . . . . . . 30
2.8 Modeling of shock-vortex interaction . . . . . . . . . . . . . . 33
2.8.1 Note on Oblique shock-vortex interaction . . . . . . 33
2.8.2 Normal shock-vortex interaction . . . . . . . . . . . 35
2.9 Interaction with multiple vortices . . . . . . . . . . . . . . . . 42
III. Shock-vortex Interaction . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Discussion on Numerical errors . . . . . . . . . . . . . . . . . 48
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 Short times . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Case 1 : Weak shock with M = 1.05 . . . . . . . . 52
s
3.3.3 Case 2 : M = 1.4 . . . . . . . . . . . . . . . . . . . 58
s
3.3.4 Case 4 : M = 1.7 . . . . . . . . . . . . . . . . . . . 63
s
3.3.5 Case 5 : M = 2.0 . . . . . . . . . . . . . . . . . . . 68
s
3.4 Decay rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Classification of the interactions . . . . . . . . . . . . . . . . 71
3.6 Details from DNS on vortex strengths and validation of the
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Inference from both Geometrical shock dynamcs(GSD) and
Euler computations . . . . . . . . . . . . . . . . . . . . . . . 76
IV. Shock surface propagation in 3D flows . . . . . . . . . . . . . . 78
4.1 GoverningEquationsofGeometricalshockdynamics(GSD)for
Shock surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 3D Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.1 A Riemann problem . . . . . . . . . . . . . . . . . . 86
4.3.2 Modeling of shock - vortex ring interaction . . . . . 87
V. Conclusions and Future work . . . . . . . . . . . . . . . . . . . . 94
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
vi
LIST OF FIGURES
Figure
1.1 Representative Scramjet Engine . . . . . . . . . . . . . . . . . . . . 2
1.2 GSD Shock propagation . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Shock oscillations shown from DNS studies . . . . . . . . . . . . . . 5
1.4 DNS results of Pirozolli and Grasso [22] showing vortices interacting
with the pulsating shock at different time intervals . . . . . . . . . . 6
2.1 Part of the net of shock locations and rays. . . . . . . . . . . . . . . 11
2.2 The point T on the shockwave has a domain of dependence defined
by the pair of characteristics TC ,TC . . . . . . . . . . . . . . . . . 13
1 2
2.3 TheRiemannproblemforGSDconsistsoffindingtheraythatemerges
from two adjacent shock segments. . . . . . . . . . . . . . . . . . . 15
2.4 The propagation of disturbance along the normal stationary shock . 18
2.5 Area Mach relations and wave-speeds in both the ξ−t plane and x-y
space as a function of m . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Shock diffraction problem setup . . . . . . . . . . . . . . . . . . . . 22
2.7 Variation of wall Mach number m with the change in diffraction
w
angle θ for two different initial shock strengths. . . . . . . . . . . . 23
w
2.8 Shock-diffractionproblemsolutionusingRoe’smodelforshockstrength
of M = 1.5 and θ = -0.2 rad. The dotted line represents the rays
O W
in fig2.8(a). The solution is plotted at intermediate time intervals . 23
vii
2.9 Relation between the change in θ with the change in strength of the
shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.10 Shock-shock problem setup. The dotted line representing the direc-
tion of propagation of the kink in the x-y space. . . . . . . . . . . . 25
2.11 VariationofwallMachnumberm withthechangeintherampangle
w
θ for two different initial shock strengths. . . . . . . . . . . . . . . 26
w
2.12 Shock-shock problem solution using Roe’s model for shock strength
of M = 1.5 and θ = 0.2 rad . The dotted line represents the rays
O W
in fig2.12(a). The solution is plotted at intermediate time intervals . 26
2.13 Transformation of the computational net for a moving medium. The
elements of the shock front are unchanged. The points A,B are
displaced to A′,B′. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.14 A shockwave moving from B to A along a solid surface. . . . . . . . 29
2.15 Schematic setup of a planar shock after reflection from corrugated
surface with strength . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.16 PropagationofasinusoidalwavefrontusingRoe’sA-Mrelation. Dot-
ted lines in 2.16(a) indicate rays. The lines with * in 2.16(b) and
2.16(c) shows the solution at final computational time . . . . . . . . 32
2.17 Changes in maximum and minimum shock strength over time . . . 32
2.18 Schematic setup of a normal shock-vortex interaction and an oblique
shock-vortex interaction . . . . . . . . . . . . . . . . . . . . . . . . 34
2.19 GSD results : At early times using Roe’s Area model for a sinusoidal
perturbation in Mach number for a planar shock of strength 1.7 . . 36
2.20 GSD results : Power law decay of the wave amplitudes with non
dimensional distance using Roe’s Area model for shock with m = 1.7 38
2.21 GSD results : Power law decay of the wave amplitudes with non
dimensional distance using Roe’s Area model for shock with m = 1.7
for larger distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.22 GSD results : At later times using Roe’s Area model . . . . . . . . 40
2.23 GSD results : At later times using Whitham’s Area model . . . . . 41
viii
2.24 GSD results : At later times using Prasad’s Area model . . . . . . . 41
2.25 GSD results : Interaction of C-E-C and E-C-E waves using Roe’s
Area model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.26 GSD results : Long range behavior of two vortex interaction . . . . 44
3.1 Tangentialvelocityprofilesforboththecompositeandisentropicvortex 46
3.2 Computational domain for the shock vortex interaction. . . . . . . . 48
3.3 Plot of enstrophy to measure numerical dissipation of the vortex . . 49
3.4 PressureconvergencewithmovingABclosertotheundisturbedshock
position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Shock vortex interaction map[13]. Hollow symbols, Grasso and Piro-
zolli [13]; filled symbols ,Ellzey etal. [10]; half-filled symbols, Inoue
and Hattori [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Weak Interaction : Time evolution of the pressure field showing the
change in the shock structure for M = 1.05 and M = 0.1 with
s v
Contour levels - 24 levels from 1.05 to 1.117 in the left side and the
monitored pressure signal on the right . . . . . . . . . . . . . . . . . 54
3.7 Time evolution of the pressure field showing Regular reflection pat-
tern in the shock structure for M = 1.05 and M = 0.5, Contour
s v
levels-36levelsfrom1.01to1.117.Theplotsontherightsideindicate
the pressure signal monitored . . . . . . . . . . . . . . . . . . . . . 55
3.8 Pressure disturbance and flow angle change monitored along AB for
shock strength M = 1.05 . . . . . . . . . . . . . . . . . . . . . . . . 56
s
3.9 Pressure disturbance and flow angle change monitored along AB for
shock strength M = 1.05 and vortex strengths M = 0.1, 0.5 and
s v
1.0 after t = 12 units. . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Power law decay of amplitudes of the pressure signal for M = 1.05. 57
s
3.11 Time evolution of the pressure field showing Mach Reflection in the
shock structure for M = 1.4 and M = 0.4, Contour levels - 64 levels
s v
from 1.01 to 2.76. The plots on the right side indicate the pressure
signal monitored . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
ix